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Terrorists' equilibrium choices when no attack method is riskless.

Introduction

Phillips (2005, 2009) has shown that the tools of financial economics, and specifically modern portfolio theory, may be applied to the analysis of terrorism. Building on this work, this paper extends the Black (1972) model of capital market equilibrium with restricted borrowing to the field of defense economics. It is shown that a capital-markets-type equilibrium can be specified for terrorist groups' choices of attack methods when those choices are made on the basis of expected returns (measured in some units) and risk (the variance of returns). Specifically, it is shown that, in equilibrium, terrorist groups will choose some combination of (1) the zero-beta combination of attack methods and (2) an equally-weighted combination of all attack methods. The zero-beta combination of attack methods is characterized by zero sensitivity to the expected returns generated by an equally-weighted combination of all available attack methods.

This paper is organized as follows. In the following section, a brief overview of the relevant literature is presented. The purpose of the literature review is to outline the common microeconomic foundations of both financial economics and defense economics and provide the background for later parts of the paper. The literature review is followed by a theoretical analysis where the salient features of the microeconomic theory and financial economics theory that is deployed in the construction of the equilibrium model of terrorist attack method choice are summarized. The theoretical analysis is complemented by an empirical analysis where the expected payoffs and characteristics of the zero-Beta combination of attack methods are computed using data extracted from the RAND Corporation's MIPT database. This is followed by a summary and conclusions.

The Literature

The analysis presented in this article is principally concerned with the behavior of terrorist agents. Terrorist agents engage in a particular form of behavior that we call terrorism. According to Enders and Sandier (2002, p. 145-146), "Terrorism is the premeditated use or threat of use of extra-normal violence or brutality by subnational groups to obtain a political, religious, or ideological objective through intimidation of a huge audience, usually not directly involved with the policy making that terrorists seek to influence." Because terrorist agents operate in a manner consistent with achieving some objective subject to constraints (see Pape (2003)), economic analysis of terrorism within the orthodox expected utility framework is permissible. (1) The analysis of the rationality of the decisions made by terrorist agents and groups is one of three streams of research identified by Faria and Arce (2005), along with the analysis of counter-terrorism policies and the analysis of the conditions that determine terrorist agent recruitment and that constitute the main themes of the extant literature dealing with the economic analysis of terrorism. The representative agent frameworks of financial economics and the stream of the defense economics literature that is concentrated on the decisions of terrorist agents share common theoretical foundations.

The treatment of agents' preferences over commodity bundles proceeds in an orthodox fashion in both financial economics and defense economics. A commodity bundle is a point in the commodity space, [[Real part].sup.l]. Given two commodity bundles, an agent expresses a preference for bundle 1 over bundle 2:

bundle 1 [??] bundle 2

Under some axioms, a utility function, which orders the points in [[Real part].sup.l], may be used to represent these preferences. A utility function, u: [[Real part].sup.l] [right arrow] [Real part], that represents a preference ordering is ordinal. The utility function only says that bundle 1 is preferred to bundle 2. It says nothing about how much more bundle 1 is preferred to bundle 2. Rational agents choose the bundle that they like best from among all the possible bundles. This is preference ordering in operation. This is expressed formally as the maximization of a utility function. This maximization usually takes place under some constraints (for example, some bundles are desirable but not affordable given the agent's budget). More importantly, this maximization usually takes place under uncertainty. Under such conditions, when the analysis involves situations where the choice set cannot be partitioned with certainty into attainable and unattainable sectors, the von-Neumann and Morgenstern (NM) class of (expected) utility functions replace the standard ordinal utility functions of elementary economic analysis.

The place for NM expected utility analysis in financial economics is clear. Assets generate (monetary or tangible) payoffs in different states and probabilities must be attached to each state. In defense economics, defining the nature of the commodity bundles is, at first, a more difficult problem. In purely theoretical work, it is sufficient to assume that terrorist agents derive utility from engaging in their behaviors (legal and illegal methods). The exact nature of the commodity bundles over which a preference ordering is defined or, in the presence of uncertainty, which form the basis for lotteries or gambles, need not be precisely defined. For example, consider the utility analysis undertaken by Enders and Sandier (2002) where a terrorist group's expected utility is

E(U) = [pi]U([W.sup.S]) + (1 - [pi])U([W.sup.F]) (1)

where: U(W) is the NM utility function and [W.sup.S] and [W.sup.F] are the net wealth equivalent measures over two states, success (S), and failure (F). In this model, the terrorist's commodity bundles are constituted by quantities of net wealth which are in turn constituted by monetary equivalents of net gains from legal and illegal activities. Such a model can be used to generate empirically testable conclusions. For example, actions by governments aimed at increasing the costs associated with a particular type of attack will reduce the relative costs of other types of attacks and produce a substitution effect. The powerful tools of expected utility analysis can be utilized to examine problems such as those confronting defense economists that may, at first, seem less than well-suited to economic analysis. Importantly, the way in which financial economists have approached the analysis of choice under uncertainty may be useful in consolidating the utility analysis of terrorism.

Mean-variance approaches have held a place of prominence in financial economics since the pioneering work of Markowitz (1952) and Tobin (1958). In financial economics, the value of mean-variance analysis--the description of risk preferences in terms of mean return (higher return increases utility) and variance (higher variance of returns decreases utility)--is its computable results (Samuelson 1970). The mean-variance approach in which an agent is assumed to rank alternatives according to a function defined over the mean and variance (or standard deviation) of the random payoff may be contrasted with the expected utility (EU) approach, where an agent ranks alternatives using expected value of a utility function defined over payoffs (Meyer 1987). Early debate on the appropriateness of the mean-variance approach focused on the consistency of any mean-variance ranking with EU ranking. It was recognized that certain additional assumptions might be required to ensure that a mean-variance ranking did not violate the NM axioms of rational behavior (Baron 1977). However, the usefulness of mean-variance analysis as a computable approximation to full EU analysis was defended vigorously, especially by Tobin (1969) (see Borch (1969), Feldstein (1969) and Tobin (1969)).

It emerged that mean-variance analysis is a special case of NM expected utility that will generate preference orderings consistent with EU (and NM axioms of rational behavior) under certain conditions: (1) when the utility function is quadratic; (2) when probabilities are compact (smaller and smaller risk) (Samuelson 1970); (3) when the payoffs of all lotteries or gambles are jointly normal such that any combination (portfolio) of lotteries will also generate payoffs that are normally distributed; and (4) when the location and scale (LS) condition is satisfied (Meyer 1987). For financial economists, conditions (1) and (2) proved troublesome. Quadratic utility functions generate "anomalous properties in the large" (Samuelson 1970, p.537) including increasing absolute risk aversion (the dollar amount invested in risky assets declines as the agent's wealth increases) and satiation. However, it is not necessary to specify a quadratic utility function (2) when using mean-variance analysis to analyse optimal decisions and, in any case, the LS condition identified by Meyer (1987) may explain the effectiveness of the theoretical and empirical results generated by mean-variance analysis of choice among combinations of lotteries (payoffs) (Levy and Markowitz 1979).

The use of mean-variance analysis to examine the optimal decisions of terrorist agents was fast explored by Phillips (2009). Mean-variance analysis can be used to examine optimal decisions without specifying a NM quadratic utility function (Baron 1977, p. 1688). In this case, an ordering of alternatives is not the point of the analysis. Rather, it is to select the optimal decision from the set of all lotteries or combinations of lotteries (portfolios). In this case, "the probability mixtures usually may be excluded from the analysis as a candidate for optimality, since every mixed strategy typically is dominated by some pure strategy and/or by a portfolio. Mean-variance analysis may then be used with an enlarged class of NM utility functions" (Baron 1977, p.1688). The propositions (3) and (4) in Baron's (1977) analysis demonstrate that, in a portfolio problem such as that examined by Phillips (2009), the mean-variance portfolio analysis is consistent with the NM axioms and a quadratic NM utility function need not be specified. The computable results generated by an analysis such as Phillips (2009) demonstrate the usefulness of the mean-variance approach.

Theoretical Analysis

Assume that the terrorist group attempts to optimize some commodity, Z, which may be called political influence. The terrorist group commits resources to attacks--classified by the RAND Corporation as: armed attacks, arson, assassination, barricade/hostage, bombing, hijacking, kidnapping, unconventional, unknown, and other--in order to obtain Z. The group thinks about the outcomes of these attacks in probabilistic terms and selects the optimal attack method (or combination of methods) on the basis of two moments (mean or expected return and variance) of the random payoff. Terrorist agents therefore make decisions on the basis of the expected amount of political influence generated from an attack and the variance of the possible divergence of the amount of political influence generated from an attack from that which was expected. (3) The utility function for the terrorist group can be stated as:

U = f([E.sub.z], [[sigma].sub.z]) (2)

The terrorist group prefers a higher mean amount of political influence, [E.sub.z], to a lower amount (more is preferred to less). The terrorist group is risk averse and prefers a lower value of [[sigma].sub.z] to a higher value. However, higher values of [[sigma].sub.z] are acceptable if they are accompanied by higher values of [E.sub.z]. Now assume Z is a function of the fatalities, F, generated by an attack method or combination of methods. Terrorist agents make decisions on the basis of the expected number of fatalities and the variance of the possible divergence of the number of fatalities from that which was expected. The utility function may therefore be stated as:

U = f([E.sub.F], [[sigma].sub.F]) (3)

The indifference curves are upward-sloping in expected return and risk space. The terrorist group chooses from the attack method combinations that constitute the choice set in order to optimize utility. This optimization is located at the point of tangency between the indifference curve and the boundary of the choice set.

The boundary of the choice set is determined by solving a quadratic programming problem. To be precise, determining the boundary of the choice set involves computing the set of combinations of attack methods with the maximum expected fatalities for a given level of risk (variance). The combinations of attack methods that fulfill this criterion are non-dominated. For each level of expected fatalities, a combination in the boundary of the choice set has the lowest risk and, for each level of risk, a combination in the boundary of the choice set has the highest expected fatalities. The details of both the portfolio formation and the relevant quadratic programming problem are explained by Phillips (2009). The terrorist group approaches the decision of selecting the optimal attack method (or combination of methods) in two steps: (1) determine the set of non-dominated (efficient) attack method combinations; and (2) choose from among this set (see Sharpe (1964, p.429)).

If all terrorist groups make their selection of the optimal decision from the set of attack method combinations on the basis of the two moments of the random payoffs, an equilibrium model of the relationship between expected fatalities and risk can be constructed. In this equilibrium model, the relationship between expected fatalities and risk is linear. Furthermore, the choices of rational terrorist groups are narrowed down to a combination of what shall be called the zero-beta combination of attack methods and an equally weighted benchmark combination. The zero-beta combination avoids the necessity for assuming the existence of a risk-less attack method. Rather, all combinations (portfolios) of attack methods in equilibrium will generate expected fatalities of an amount equal to that generated by the zero-beta combination plus some premium in the form of additional fatalities. The equilibrium equation towards which the analysis will work is presented simply as follows:

E([F.sub.p]) = E([F.sub.zero]) + [[beta].sub.P](E([F.sub.e]) - E([F.sub.z])) (4)

where: E([F.sub.zero]) is the expected fatalities generated by the zero-beta combination of attack methods; [[beta].sub.P], measures the sensitivity of a combination of attack methods to a combination constituted by all attack methods (equally weighted); and E([F.sub.e]) is the expected fatalities from this equally weighted combination. Equation 4 is an equilibrium relationship that holds for all attack methods and combinations of attack methods. This is analogous to the Capital Asset Pricing Model of Sharpe (1964), Lintner (1965) and, especially, Black (1972) that holds such a place of prominence in financial economics. Interestingly, though, some of the assumptions that underlie the model are less troublesome when applied analogously to terrorist behavior than in their traditional application to investor behavior.

Assumption 1: The common probability distribution describing the random payoffs (fatalities) of each attack method is joint normal (or joint stable with a single characteristic exponent) (Black 1972, p.444); or, failing this, the LS condition holds (Meyer 1987).

Assumption 2: No attack method is without risk. No attack method generates expected fatalities with zero variance.

Assumption 3: Terrorist groups choose attack method combinations to maximize fatalities and, in so doing, their end-of-period holdings of Z (defined earlier).

Assumption 4: Terrorist groups are risk averse. This does not rule out any attack method or combination. It simply requires that increased risks are appropriately rewarded. A risk averse terrorist agent may be indifferent between a very risky attack method that yields a high expected payoff and a very low risk attack method that yields a low expected payoff.

Assumption 4(a): It might additionally be assumed that terrorist groups display decreasing absolute and constant relative risk aversion (DARA and CRRA, respectively). DARA and CRRA imply that the terrorist group will allocate larger absolute amounts of resources towards attacks as their holdings of Z increases but these resource allocations will represent a constant percentage of their total resources. (4)

Assumption 5: A terrorist group may construct a combination of attack methods in which the weightings assigned to any of the risky attack methods may be of any size (positive or negative). For example, hijacking may be accorded a weighting of -10% and the total weighting of a combination may exceed 100%. (5)

There is, from this point, a very simple way to derive the equilibrium model. This is the derivation proceeds along the lines outlined by Sharpe (1964) and Black (1972). However, something must first be said about assumption 5. Of course, such a weighting scheme is unrealistic--although negative weightings for illegal (attack) methods could be interpreted as legal activities (6)--but it merely simplifies the derivation. Lintner (1971) showed that equilibrium is unaffected when negative weightings are disallowed and, indeed, it is obvious that in equilibrium no terrorist group will hold a combination of attack methods where some attack methods are assigned negative weightings. Allowing negative weightings, therefore, cannot alter the equilibrium. The other problem (of positive weightings of any size) is equally easy to overcome. A combination of attack methods where the sum of the weightings exceeds 100% is formed by taking a negative position in the zero-beta combination and transferring those resources to the optimal risky combination of attack methods. Since, in aggregate, the net position of all terrorist agents in the zero-beta combination must necessarily be zero (this is explained later), the equilibrium is again unaltered by assumption 5.

The efficient (non-dominated) set of attack method combinations derived as in Phillips (2009) may, under assumptions one to five above be represented as a combination of two combinations. The weights of each individual attack method within these two combinations may be arranged to generate any of the combinations in the efficient set. One convenient statistic that may be utilized to shortcut the complexities of the underlying quadratic programming problem and drive to the heart of the structure of the correlation of fatalities generated by various attack methods is a statistic that shall be called Beta. The Beta, [[beta].sub.i], for any given attack method or combination ([[beta].sub.P]) is a measure of the sensitivity of its expected payoffs to the expected payoffs of an equally weighted combination of all attack methods. Obviously, the equally weighted combination must have a Beta equal to one (because it co-varies perfectly with itself). Beta is the salient measure of risk because all of the risk associated with the variance of individual attack methods may be removed by combining the non-perfectly correlated attack methods. The only risk that will remain is the risk associated with each combination's sensitivity to the payoffs generated by the equally weighted combination.

Imagine two combinations, A and B, of attack methods. Positive combinations of A and B lay on the line segment that connects them. In expected payoff-Beta space, combinations of any two risky combinations of attack methods lie on the straight line connecting them. The proof is simple. The expected payoff of a combination, D, of combination A and combination B is

E([F.sub.P,D]) = xE([F.sub.P,A]) + (1 - x)E([F.sub.P,B]) (5)

The equation for the Beta coefficient that measures the sensitivity of combination D to the equally weighted combination is the weighted average of the Beta coefficients of combination A and combination B

[[beta].sub.P,D] = x[[beta].sub.P,A] + (1 - x)[[beta].sub.P,B] (6)

Following Elton et al. (2003, p.296), solve for x in Eq. 6 and substitute into Eq. 5 to yield an equation for a straight line

E([F.sub.P]) = a + b[[beta].sub.P] (7)

Now assume that there is a third combination, C, that lies directly above combination A. Combination C, therefore, has superior expected payoff but the same level of risk. Of course, terrorist groups would allocate resources to C rather than A until the amount fatalities that could be expected from C is equal to A (either because the increasing prevalence of C leads to security responses that reduce its payoffs or because of a more sophisticated price adjustment, where prices are defined in a manner similar to the definition sketched by Phillips (2005)). Within the context set down by the assumptions (above), the terrorist group could use a negative weighting in A to transfer resources to C. The result is a positive payoff with no risk. This situation is not permissible in equilibrium. Therefore, if combinations of attack methods lie on a straight line, in equilibrium all combinations of attack methods must lie on the same straight line (because if they did not, it would be possible to construct a combination of attack methods that earned superior payoffs with no risk). The equation for this equilibrium (straight line) relationship is, of course, Eq. 4. The derivation and proof is relatively straightforward. The following is a cut down version of Black's (1972, pp.446-450) derivation and relies on the presentation of Elton et al. (2003, pp.311-312).

The efficient (non-dominated) set of all attack method combinations may be written as a combination of two basic combinations p and q. The proof of this statement is presented by Black (1972) and culminates in his Eq. 12. Because p and q are not unique, it is possible to transform these basic combinations into different combinations u and v, which are combinations of combinations p and q that solve Black's (1972) Eq. 12. All efficient combinations of attack methods can be formed from u and v (although u and v need not be efficient themselves) (Black 1972, p.449). If the objective is to search for the equilibrium straight-line relationship in expected payoff-Beta space, two points are needed. The most obvious two points are (1) the point that characterizes the combination of the attack methods that has a zero Beta (the intercept), and (2) the point that characterizes the combination of attack methods that has a Beta of 1 (the equally weighted combination). If it is possible to generate every efficient combination from a weighted combination of combinations p and q, then p and q must have different Betas. It must be possible to form combinations u and v with arbitrary Betas by choosing appropriate weights (Black 1972, p.449). So the combinations with Beta 0 and Beta 1 are constructed.

Whereas the efficient (non-dominated) combinations of attack methods that are computed using the quadratic programming techniques (see Phillips (2009)) are a convex set in expected payoff-variance space, the steps taken in the analysis presented above generate an equilibrium relationship that may be depicted geometrically as a straight line in expected payoff-Beta space. This exploits the correlation structure of the expected payoffs of different attack methods and combinations of attack methods. The conclusion of the theoretical part of this paper may be stated as follows. In equilibrium, terrorist groups will form efficient (nondominated) attack method combinations by choosing weighting schemes for two basic combinations: (1) the zero-Beta combination; and (2) the equally weighted combination. All terrorist groups' attack method combinations will be combinations of these two basic combinations and all terrorist groups' attack method combinations will lie on the straight line that connects the zero-Beta and the equally weighted combinations in expected payoff-Beta space.

Empirical Analysis

Two points in expected payoff-Beta space are required: (1) the point that represents the equally weighted (benchmark) combination of attack methods; and (2) the point that represents the zero-Beta combination of attack methods (the intercept). The zero-Beta combination of attack methods will have zero sensitivity to the variation in the expected payoffs generated by the equally weighted combination. Of course, the equally weighted combination co-varies perfectly with itself and has a Beta of 1. The equally weighted combination is, in fact, a proxy for a complete combination of all attack methods. In the purely theoretical work of financial economics, the market portfolio is assumed to contain all marketable assets (including artwork and human capital). In the empirical work of financial economics, however, a benchmark portfolio (usually a market index such as the S&P500) is deployed as the proxy for the market portfolio. In the analysis presented in this section, the equally weighted combination of the attack methods identified by the RAND Corporation serves the purpose of a proxy for the entire collection of possible attack methods. (7) In practice, a decision must always be made on the constitution of such a proxy. A more elaborate construct than the equally weighted combination suggested here is certainly possible. This task, too, is left for future research. (8)

The characteristics of the two combinations (zero-Beta and equally weighted) are determined through an application of the portfolio methods detailed by Phillips (2009). The RAND Corporation's data is utilized as the basis for the analysis. This data is sourced from the RAND Corporation's MIPT database (which has now been replaced by the RDWTI) for the period 1968 to 2006. Unlike Phillips (2009), the focus in this analysis is on fatalities only. Injuries are not considered. In order to facilitate the analysis, the raw data must be transformed into a payoffs series that reflects the fatalities generated each year by each attack method (weighted by the number of attacks). The payoffs series that is produced reflects the payoff per attack per year (for each attack method). This provides a more meaningful basis for the analysis and one that is much more suitable for statistical analysis (stationarity is more likely and the distribution of payoffs more likely to be approximately normally distributed). The mean, standard deviation and variance for each attack method is presented in Table 1. These descriptive statistics may be interpreted as: (1) the average payoff (measured in fatalities) per attack per year over the 38 year period 1968 to 2006; and (2), the variability of these payoffs.

In order to derive the two combinations (zero-Beta and equally weighted), a weighting scheme is applied to the payoffs time series. Terrorist groups construct a combination of attack methods in which each of the attack methods is assigned the same weighting. Given that there are ten attack methods classified by the RAND Corporation, an equally weighted combination of attack methods involves the assignment of a weighting of 10%, or 0.10, to each attack method. This combination of attack methods will generate a particular (expected or mean) payoff (measured in fatalities) per attack per year and will be characterized by a particular level of risk (variance or standard deviation). The expected payoff of the equally weighted combination of attack methods is relatively straightforward to compute. However, the risk (variance) of the combination necessitates the consideration of the covariance that characterizes the interactions between the payoffs of each attack method in the combination. The calculations reveal that the equally weighted combination has expected payoff 1.8116 (fatalities per attack per year) and risk (standard deviation of generated payoffs) 2.71%.

In expected payoff-Beta space, the equally weighted combination of attack methods has an expected payoff of 1.8116 fatalities and Beta 1. The task of determining the composition of the zero-Beta combination is slightly more complicated. A combination of attack methods with zero-Beta will have no sensitivity to the equally weighted benchmark combination. Beta can be computed using Eq. 8:

[[beta].sub.i] = [[sigma].sub.i,equal-weighted]/[[sigma].sup.2.sub.equal-weighted] (8)

The numerator in Eq. 8 is the covariance of the payoffs of a combination of attack methods with the payoffs of the equally weighted combination of attack methods and the denominator is the variance of the payoffs of the equally weighted combination. A Beta of zero will be generated by a combination of attack methods that generates a series of payoffs that has a covariance of zero with the equally weighted benchmark combination. Because there may be more than one solution, it is necessary to find (1) a combination that yields zero covariance with the equally weighted combination; and (2) has the minimum variance among the set of possible solutions. The combination that yields a series of payoffs that produces a covariance of zero with the payoffs produced by the equally weighted combination (for the period 1968 to 2006) and has the minimum variance among the set of possible solutions is the combination derived from the weighting scheme presented in Table 2. The expected payoff generated by the zero-Beta combination is 0.4157 fatalities per attack per year. This is a payoff generated by a combination that exhibits no systematic covariance with the equally weighted benchmark combination.

In equilibrium, each terrorist group will hold a combination of attack methods that is a combination of (1) the zero-Beta combination, and (2) the equally weighted benchmark combination. The exact combination held by each individual terrorist group will be a function of that group's risk aversion. The equilibrium model has the advantage of yielding a two combination solution. All terrorist groups, regardless of their precise level of risk aversion, can be satisfied by combinations formed from combinations of the zero-Beta combination and the equally weighted combination. More risk-averse terrorist groups will hold a combination with a heavier weighting assigned to the zero-Beta combination. Less risk-averse terrorist groups will hold a combination of attack methods with a heavier weighting assigned to the equally weighted combination. In equilibrium, of course, no negative weightings will be assigned to any particular attack method or combination of attack methods. In equilibrium, the aggregate holdings of the zero-Beta combination will sum to zero. In equilibrium, terrorist groups, in aggregate, hold the benchmark combination of attack methods. Of course, since defense economists will mainly be interested in particular terrorist groups, it is essential to understand the make-up of the two portfolios (zero-Beta and equally weighted) from which all choices of combinations of attack methods can be satisfied.

Conclusions

The mean-variance analysis of the choice set and the task of selecting the optimal decision from the choice set may be analyzed in the manner outlined by Phillips (2009). Taking this as the starting point and assuming that all agents makes their decisions on the basis of the two moments of the random payoffs generated by attack methods and combinations of attack methods, an equilibrium relationship between expected payoffs and risk can be constructed. The properties of this equilibrium relationship are such as to permit the identification of the two base combinations from which all combinations of attack methods will be formed in equilibrium. It is shown that the relationship between the expected payoff and risk (Beta) in equilibrium is a straight line in expected payoff-Beta space. All combinations of attack methods chosen by the terrorist groups will be combinations of two base combinations: (1) the zero-Beta combination; and (2) the equally weighted combination.

The computed results show that the expected-payoffs of the two efficient attack method combinations are low. Only 1.8116 fatalities per attack are expected from the equally weighted benchmark combination and 0.4157 fatalities per attack are expected to be generated by the zero-Beta combination. The terrorist groups face a risk-return trade-off. A higher Beta risk is associated with a higher expected payoff and a lower Beta risk is associated with a lower expected payoff. The weightings of the equally weighted benchmark combination are, of course, quite straightforward. The weightings of the zero-Beta combination computed as part of the empirical analysis ate more interesting. The zero-Beta combination, which is one of the two combinations from which all combinations selected by the terrorist groups will be formed, assigns heavy weightings to arson, assassination and kidnapping. The lowest risk combination assigns negligible weightings to the other attack methods. Heavily risk-averse terrorist groups may therefore be expected to have a tendency toward arson, assassination and kidnapping. The less risk-averse the terrorist groups happen to be, the more their selection of the optimal decision from the choice set will drift toward the equal weighted combination. Significantly, the transfer of resources from more risk-averse groups to less risk-averse groups may permit the latter to select a combination of attack methods that heavily weighs the most risky but also the most damaging attack methods. This is certainly an important topic for future research.

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P. J. Phillips ([mail])

School of Accounting, Economics and Finance, University of Southern Queensland, Toowoomba,

QLD 4350, Australia

e-mail: phillipsp@usq.edu.au

(1) Following a similar approach to Becker (1968), many studies of terrorist behavior make use of game theoretic or utility theoretic frameworks (see Landes (1978), Sandler et al. (1983), Frey and Luechinger (2003), Sandier and Arce (2003), Arce and Sandier (2005), and Sandier and Arce (2007)).

(2) However unattractive quadratic utility may be to financial economists, defense economists must consider the plausibility of such functions (which ensure the NM axioms are not violated in mean-variance preference orderings) in the context of terrorist behavior. Of special interest are points where the utility function vanishes (to a single point), which may be relevant to the study of suicide terrorism.

(3) Unlike Phillips (2009), this analysis considers only fatalities and not fatalities + injuries.

(4) There is, however, no need to specify the exact type of absolute and relative risk aversion. The assumption of risk aversion is all that is necessary.

(5) A terrorist group may form a combination where the sum of the weightings of the individual attack methods that constitute the combination exceeds 100%.

(6) This is explored in a future paper.

(7) All interesting approach might be to analyze the expected returns to terrorist groups' assets (including suicide bombers, equipment, financial securities, cash, etc) rather than the payoffs of attack methods.

(8) It should be noted, however, that an equally weighted combination probably adequately reflects the systematic risk of terrorism.

DOI 10.1007/s11293-010-9253-z

Published online: 31 December 2010

[c] International Atlantic Economic Society 2010
Table 1 The risk and return of terrorist attack methods

                            Standard    Fatalities per
                 Variance   deviation   attack per year

Armed attacks       1.26       1.12          1.30
Arson               0.57       0.75          0.32
Assassination       0.15       0.39          1.04
Hostage           135.80      11.65          3.62
Bombing            28.31       5.32          4.60
Hijacking          14.82       3.85          1.57
Kidnapping          0.11       0.34          0.39
Other               3.76       1.94          0.47
Unconventional    576.28      24.01          3.88
Unknown            16.03       4.00          0.92

Table 2 Weightings of attack methods in the zero beta combination

Attack method    Weighting in zero beta combination

Armed attacks                  -0.017
Arson                           0.2760
Assassination                   0.3011
Hostage                        -0.012
Bombing                        -0.023
Hijacking                       0.0056
Kidnapping                      0.4726
Other                          -0.005
Unconventional                 -0.002
Unknown                         0.0056
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Author:Phillips, Peter J.
Publication:Atlantic Economic Journal
Geographic Code:1USA
Date:Jun 1, 2011
Words:6095
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